COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Coupled Vibration Analysis of Multiple Launch Rocket System by Finite
Element Method
Bingshang Li, Xinqi Xu *
Naval Aeronautical Engineering Institute, Yantai, Shandong, 264001 China
Email: [email protected]
Abstract In multiple launch rocket system (MLS), initial disturbing is one important factor to influence the firing
precision. It is meaningful theoretically and valuable practically to Study and to take right theory and effective
calculating methods to analyze the vibration of MLS. Based on Constructive Solid Geometry (CSG) and Finite
Element Method (FEM), the elastic MLS modal is built and the coupled vibration response of the system when
firing constantly is simulated. In order to verify the availability of researching methods, the conclusions are
compared to practical situation. The result provides a reference in MLS design.
Key words: multiple launching rocket system; coupled vibration; finite element method
INTRODUCTION
For one MLS, there is a strong wallop on it when one rocket is fired (Fig.1). Then the MLS will be in a condition of
being vibrated and the vibration will lasts for a time. In this situation, if rockets are fired constantly, the latter will be
launched under the condition of MLS’ being vibrated. Obviously, the wallop caused by the former one will influence
the launching precision of the latter. That is the problem. And in the paper, it is called the coupled vibration problem of
MLS when firing constantly. The simulation to such problem does good to judge the emitter which is transfigured least
and then provides reference to decide which rocket should be fired next time.
Figure 1: Coupled vibration of MLS when firing constantly
FINITE ELEMENT PROCEDURES IN STRUCTURE DYNAMIC ANALYSIS
Structure dynamic analysis is calculating the Structure dynamic response under dynamic loads. There are four steps in
finite element procedures in structure dynamic analysis. First, the solid structure is meshed. Second, the kinetic energy
function ( T ( e ) ), potential energy function ( W ( e ) ) and consuming energy function ( R ( e ) ) of each element are formed.
Third, the T ( e ) , W ( e ) and R ( e ) of all elements are assembled to get the T , W and R of the system.
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1 & T
1 & T
⎧
(e)
(e) &
&
⎪ T = ΣT = 2 ( Δ ) ( ΣM ) Δ = 2 ( Δ ) M Δ
⎪
T
T
T
T
1
1
⎪
(e)
(e)
(e)
⎨W = Σ W = ( Δ& ) ( Σ K ) Δ& − ( Δ& ) ( Σ Q ) = ( Δ& ) K Δ& − ( Δ& ) Q
2
2
⎪
T
T
1
1
⎪
(e)
(e) &
&
&
&
⎪ R = ΣW = 2 ( Δ ) ( ΣC ) Δ = 2 ( Δ ) C Δ
⎩
(1)
where:
M ( e ) —— mass matrix of element, M ( e ) = ∫ ρ N T NdV ,
Ve
K ( e ) —— rigid matrix of element, K ( e ) = ∫ BT DBdV ,
Ve
C ( e ) —— damped matrix of element, Δ& C ( e ) = ∫ μ N T NdV ,
Ve
(e)
Q —— loading vector of element,
Δ —— displacement vector of node,
Fourth, put equation (1) into Lagrange equation
⎧ d ∂L ∂L ∂R
−
+
=0
⎪
⎨ t ∂Δ& ∂Δ ∂Δ&
⎪⎩ L = T − W
(2)
Then, the dynamic equation of the whole structure is got in FEM
&& (t ) + C Δ& (t ) + K Δ (t ) = Q(t )
MΔ
(3)
To solve equation (3), the displacement, the speed and the acceleration responding of all nodes can be obtained.
Through transfer matrix, the dynamic stress and dynamic strain can also be obtained.
MODELING AND LOADING
1. Modeling Based on related references, one MLS which include the emitter, the strengthened frame, the pitching
device, the turning device and the radix device are modeled. The radix device is made of alloy steel and others are
made of aluminum alloy. With the Solid element, the model is meshed which include 67605 elements and 167949
nodes by finite element. The DOF value of all nodes in the lower area of radix device is 0. The effect is just as Figure.2.
Figure 2: Finite element model of the MLS
2. Loading Based on related references, such assumptions are made [3-5]:
(1) The transmit rails are straight;
(2) The emitter is short and its weight is less much than the whole weight of MLS;
(3) The influences of environment (such as loads of wind, movements of carrier) are ignored.
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(4) The MLS is firing and the pitching device and the turning device are all locked. There is no relative movement
among organs.
(5) When the rocket is moving in the emitter, the friction between the rocket and the emitter, the inertia of landscape
orientation and the changes of loading location (such as gravity of missile, gas-fired impulsion) are all ignored.
Then, the loads when rockets being launching constantly are stepped. And the loads when simulating include gravity
of rocket, gas-fired impulsion and closing force. Table 1 is about the Load Step Opts when first rocket is being fired
Table 1 Load Step Opts when first rocket is being fired
Step
number
Load
Stepped or
ramped
Automatic time
stepping
Time at end of
load step(seconds)
1
gravity of eight rocket(on)
ramped
on
6
2
no load
on
20
3
closing force(on)
ramped
on
20.05
4
closing force(off)
stepped
on
20.11
5
gravity of first rocket(off)
stepped
on
20.111
6
gas-fired impulsion(on)
ramped
on
20.116
7
gas-fired impulsion(off)
ramped
on
20.121
8
no load
on
23.00
1
SIMULATING CONDITIONS
2
3
4
6
7
8
5
(1) Emitters’ numbers (Fig. 3)
Figure 3: Emitters’ numbers
(2) Firing order
6——7——5——8——2——3——1——4
(3) Emitters’ obliquity 30 o
(4) Firing interval 3s
(5) Position of sampling points Each sampling point is 360 mm or 1200 mm apart from the front of emitter.
There are sixteen sampling points all.
(6) Rules to analyze the results of simulating The rule that displacement changed with time on X orientation is in
proportion to the disturbing to the angle of orientation. And the rule that displacement changed with time on Y
orientation is in proportion to the disturbing to pitching angle. The rule that displacement changed with time on Z
orientation in proportion to the disturbing to the rockets’ flying ahead.
SIMULAING RESULT
1. Vibration responding curves The curves are shown in Figs. 4-11, where the lines denoted by UX, UY and UZ
are the vibration curves of displacement on X, Y and Z direction, respectively:
(1) Vibration responding curves of Node 41220 and Node 41743 of Emitter 1 (Fig. 4);
(2) Vibration responding curves of Node 27540 and Node 28063 of Emitter 2 (Fig. 5);
(3) Vibration responding curves of Node 13860 and Node 14361 of Emitter 3 (Fig. 6);
(4) Vibration responding curves of Node 180 and Node 703 of Emitter 4 (Fig. 7).
(5) Vibration responding curves of Node 48060 and Node 48583 of Emitter 5: (Fig. 8):
(6) Vibration responding curves of Node 34380 and Node 34903 of Emitter 6: (Fig. 9):
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(7) Vibration responding curves of Node 20700 and Node 21201 of Emitter 7: (Fig. 10):
(8) Vibration responding curves of Node 7020 and Node 7543 of Emitter 8: (Fig. 11):
Figure 4: Vibration responding curves of Node 41220 and Node 41743 of Emitter 1
Figure 5: Vibration responding curves of Node 27540 and Node 28063 of Emitter 2
Figure 6: Vibration responding curves of Node 13860 and Node 14361 of Emitter 3
Figure 7: Vibration responding curves of Node 180 and Node 703 of Emitter 4
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Figure 8: Vibration responding curves of Node 48060 and Node 48583 of Emitter 5
Figure 9: Vibration responding curves of Node 34380 and Node 34903 of Emitter 6
Figure 10: Vibration responding curves of Node 20700 and Node 21201 of Emitter 7
Figure 11: Vibration responding curves of Node 7020 and Node 7543 of Emitter 8
2. Displacement-deformed shape of MLS The displacement-deformed shape of MLS is shown in Fig. 12.
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Figure 12: Displacement-deformed shape of MLS on three orientations (X, Y, Z)
CONCLUSION
(1) From Figure 4 to Figure 11, it can be observed that the MLS vibration is more acute in 26-29-32 and 38-41-44
seconds than in other seconds. And the time is just when the rocket is being launching from the emitter outside. So it
can be reasoned that the vibration acute more when being launched outside. The torque (referring to center axis of the
MLS) when being launched outside is much greater than inside.
(2) From Figure 4 to Figure 11, it can be observed that the vibration is most acute in Z orientation and less in Y
orientation and least in X orientation. The reason is that the component of force (gas-fired impulsion and closing force)
in Z orientation is much greater than the component in Y orientation and there is no force in X orientation when the
obliquity of emitters is 30 degrees. It can also be reasoned that the vibration acuteness is related to the obliquity of
emitters and that the vibration in Y orientation will be more acute than in Z orientation.
(3) From Figure 4 to Figure 11, it can be observed that the slop of vibration responding curve when gas-fired impulsion
and closing force are loading on the MLS varies much greater than when gravity of missile is loading on the MLS. The
reason is that gas-fired impulsion and closing force is greater much more ten times than gravity of missile in fact.
(4) The figure 12 is the displacement-deformed shape of MLS when launching on forty-four second. And the color
varieties are in proportion to the displacement varieties. From the figure12, it can be visually observed that the eight
emitters of MLS varies in displacement in three orientations (X, Y, Z). Such varieties all can influent the rocket initial
flight stance.
REFERENCES
1. Zeng P. Finite Element Analysis and Applications. Tsinghua University Press, Beijing, China, 2004 (in
Chinese).
2. Rao SS. The Finite Element Method in Engineering. Pergamon Press, 1982.
3. Rui XT, Wang GP, Lu YQ. Advances in rocket launch dynamics. 20th International Symposium on Ballistics,
DEStech Publications, Pennsylvania, USA, 2004, pp. 408-415.
4. Rui XT, Lu YQ. Simulation and Test Methods of Launch Dynamics of multiple launch rocket system. National
Defense Industry Press, Beijing, China, 2003 (in Chinese).
5. Wang GP, Rui XT, Chen WD. Dispersion Simulation Technique of Multiple Tube Weapon. Journal of system
simulation, 2004; 16(5): 963-966.
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